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\title[Crossover w/ 3-species]{BEC-BCS Crossover with the Feshbach Resonance for a Three-Hyperfine-Species Model}
\author[Guojun Zhu]{Guojun Zhu}
\institute{University of Illinois at Urbana-Champaign}
\date{Dec. 15th, 2011}
\pgfdeclareimage[height=1cm]{university-logo}{uiucLogo2.jpg}
\logo{\pgfuseimage{university-logo}}
%\usetheme[language=english,
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%]{TorinoTh}
%\rel{Anthony J. Leggett}
%\usetheme[hideothersubsections, width=2cm]{PaloAlto}
\usetheme[hideothersubsections]{Berkeley}

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\AtBeginSection[]
{
  \begin{frame}
    \frametitle{Table of Contents}
    \tableofcontents[currentsection]
  \end{frame}
}

\begin{document}

\frame{
%\frametitle{Dissertation}
\titlepage}

\section[Outline]{}
\frame{
\frametitle{Outline}
\tableofcontents
}

\section[Problem]{Three-species narrow Feshbach resonance}\label{sec:intro:one}
%\frame{
%
%\frametitle{Single-atom Hamiltonian and two-atom interaction}
%
%\begin{itemize}[<+->]
%\item Single atom spin-part Hamiltonian:
%\begin{equation}
%H_{spin}=A \mathbf{I}\cdot\mathbf{S}-\mu_{e}\mathbf{B}\cdot\mathbf{S}-{\mu}_{n}\mathbf{B}\cdot\mathbf{I}
%\end{equation}
%One level splits into a few hyperfine spins, marked by  $(F,F_{z})$.  
%$\mathbf{F}=\mathbf{S}+\mathbf{I}$
%
%
%\item Channels: one  configuration of hyperfine spins for one atom pair in interaction, $|{F_{ }^{(1)},F_{z}^{(1)}}\rangle\otimes|{F_{ }^{(2)},F_{z}^{(2)}}\rangle$ 
%\item Interaction mostly due to electrons
%\begin{equation}\label{eq:intro:two}
%V=f(r)+g(r)\mathbf{S_{1}}\cdot\mathbf{S_{2}}
%\end{equation}
%\item Short-range potential \& low-energy:  characterized with single parameter, s-wave scattering length, $a_{s}$
%
%\item Channels are mixed.  Resonance are possible. 
%\end{itemize}
%}



\frame{
 \frametitle{Dilute ultracold alkali gas}
 \begin{itemize}[<+->]
\item


 Dilute ultraold alkali gas: \myemph{short-range potential} \& \myemph{low-energy}:  characterized with a \myemph{single parameter}, s-wave scattering length, \myemph{ $a_{s}$}
 \item
 Feshbach resonance: effective open-channel s-wave scattering length
\begin{equation}
a_{s}(B)=a_{bg}\br{1+\frac{\Delta{B}}{B-B_{0}}}
\end{equation}
 \begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.6\textwidth]{FeshbachAs}
%\caption{S-wave scattering length in Feshbach resonance} 
\label{fig:intro:Feshbach}

\end{center}
\end{figure}
\end{itemize}

 }
 
 
 \frame{
 \frametitle{Potentials in a Feshbach resonance} 
  Channel: one  configuration of hyperfine spins for \myemph{one atom pair} in interaction, $|{F_{ }^{(1)},F_{z}^{(1)}}\rangle\otimes|{F_{ }^{(2)},F_{z}^{(2)}}\rangle$ 

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.8\textwidth]{2potential}

%Levels are marked with $F$ and $F_{z}$ {(see Footnote \ref{foot:intro:f} in page \pageref{foot:intro:f})} %Note that the energy of the $\ket{F=\nth{2},F_z=-\nth{2}}$ state first increases with the magnetic field, $B$, at low field then decrease at high field. In the same way, the energy of the $\ket{F=\frac{3}{2},F_z=-\nth{2}}$ state first decreases and then increases with the magnetic field.  
\label{fig:intro:li6}
\end{center}
\end{figure}
}


 \frame{
\frametitle{Scattering state and the bound state in a Feshbach resonance}
%\begin{columns}
%
%\begin{column}{0.6\textwidth}

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.6\textwidth]{levels}
%\caption{Energy levels in a Feshbach resonance\label{fig:intro:levels}} 
\end{center}
\end{figure}
Resonant point is not the level crossing point. The shift $\mathcal{K}$ is large than most energy scales.
$\delta=\tilde\delta-\mathcal{K}$

\pause
\begin{tabular}{|p{4cm}|p{4cm}|}
\hline
 $\mathemph{\abs{\delta}\ll\delta_{c}}$&$\mathemph{\abs{\delta}\gg\delta_{c}}$\\\hline
 Atoms are predominantly in the open-channel.&Atoms are predominantly in the closed-channel.\\
 \hline
\end{tabular}
%\end{column}
%\begin{column}{0.35\textwidth}

%\end{column}
%\end{columns}
}


\begin{frame}
\frametitle{Broad and narrow Feshbach resonance }

   \myemph{Many-body} introduce an important scale: 
         \myemph{ $E_F$}
         
 \pause
       \myemph{Broad resonance}: $\mathemph{\delta_c\gg{}E_F}$
        \begin{itemize}

\item The closed channel weight is negligible everywhere.
\item The closed channel is just like a simple knob for $a_{s}$
\item \myemph{The solutions of the single-channel model   applies.}
\item Most literatures  integrate the closed-channel out at broad resonance limit. 
\end{itemize}
   

              \pause
 
       \myemph{Narrow resonance}:  $\mathemph{\delta_c\ll{}E_F}$
        \begin{itemize}

\item The closed channel weight is substantial or even dominant close to the resonance.
\item \myemph{A two-channel  many-body model is necessary}. 
\item Studied by Gurarie and Radzihovsky \footnote{[Gurarie and Radzihovsky, Annals of Physics, 322(1):2 � 119.  2007]}
\end{itemize}
\end{frame}

\frame{
\frametitle{Three species problem}

\begin{itemize}
\item <1-> Two channels might share a \myemph{common} hyperfine species.
\item <2-> Interchannel Pauli exclusion between two channels.
\item <2->No counterpart in the two-body physics.  
\item <3->Only important in the narrow resonance. 
\item <3->No study on this subject in the literature. 
\item <1->Example: the resonance close to 834G in $^{6}$Li, 
\begin{itemize}
\item open-channel: $\ket{\frac{1}{2},-\nth{2}}\otimes\mathemph{\ket{\nth{2},+\nth{2}}}$;
\item closed-channel: $\ket{\frac{3}{2},-\nth{2}}\otimes\mathemph{\ket{\nth{2},+\nth{2}}}$.
\end{itemize}    It is a broad resonance though. 

\end{itemize}
}

\section[Single-channel]{Single-channel crossover}
\frame{
\frametitle{Single-channel crossover}
\begin{itemize}
\item  $\prod_{\vk}(u_{\vk}+v_{\vk}a^{\dg}_{\uparrow\vk}a^{\dg}_{\downarrow-\vk})\ket{0}=A\exp{}(\sum_{\vk}\frac{v_{\vk}}{u_{\vk}}a^{\dg}_{\uparrow\vk}a^{\dg}_{\downarrow-\vk})\ket{0}$
\item Continuously varying interactions ($a_{s}$)
\end{itemize}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.7\textwidth]{crossover}
%\caption{Energy levels in a Feshbach resonance\label{fig:intro:levels}} 
\end{center}
\end{figure}
}
\frame{
\frametitle{The gap equation and the number equation}
 \myemph{The renormalized gap equation}
\begin{equation}\label{eq:pathInt:gapRenormalized}
\boxed{-\frac{m\mathcal{V}_{0}}{4\pi{}a_{s}}=\sum_{\vk}\mbr{\frac{1}{2E_k}-\nth{2\epsilon_{\vk}}}}
\end{equation}
$E_\vk=\sqrt{(\epsilon_\vk-\mu)^2+\abs{\Delta_0}^2}$;  $\mathcal{V}_0$: total volume.

\myemph{The number equation}
\begin{equation}
\boxed{N=\nth{\mathcal{V}_0}\sum_{\vk}\mbr{1-\frac{\epsilon_{\vk}}{E_{\vk}}}}
\end{equation}

}

\frame{
\frametitle{$\mu$ and $\Delta$ in the crossover}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.8\textwidth]{SingleChannelCrossoverMuDelta}
\caption{The chemical potential $\mu$ and gap $\Delta$ in the mean field level over crossover} 
\label{fig:pathInt:meanField}
{\small All quantities in the unit of energy ($\mu$, $\Delta$) are rescaled with the Fermi energy $E_{F}$ and the s-wave scattering length $a_{s}$ is rescaled with $1/k_{F}$.  }
\end{center}
\end{figure}
}


\section[Two-channel]{Two-channel three-species crossover model}

\frame{
\frametitle{Extremely narrow resonance}
The coupling between two channels is assumed to be extremely weak ($Y\to0$). $\mathcal{K}\to0$, $\delta_{c}\to0$. Infinitely narrow resonance.  Two channels share the same chemical potentials. 
\begin{figure}[hhtb]
	\centering
	         \subfloat[$E_{F}<\tilde\delta$]{\label{fig:narrowFR:aboveSea}\includegraphics[width=.2\textwidth]{narrowFRabove.eps}}\quad
		\subfloat[$0<\tilde\delta<E_{F}$]{\includegraphics[width=.30\textwidth]{narrowFRin.eps}\label{fig:narrowFR:inSea}}\quad
		\subfloat[$\tilde\delta<0$]{\label{fig:narrowFR:belowSea}\includegraphics[width=.20\textwidth]{narrowFRbelow.eps}}
	%\caption{Extremely narrow resonance\label{fig:narrowFR}}
	%\small{The shaded area is occupied by atoms. }
	%\parbox{0.7\textwidth}{\small{  In Fig. \subref{fig:narrowFR:inSea} chemical potential would be close to the closed-channel bound state level (besides small shift due to the open-channel intra-channel coupling) and the ``Fermi sea'' above is empty. }}
\end{figure}
\begin{itemize}
\item Energy is counted from Fermi surface;
\item Atoms distributed in both channels in (b); 
\item The inter-channel Pauli exclusion persists in (b).
\end{itemize}
}

\frame{
\frametitle{Interchannel Pauli exclusion: handwave discussion}
\begin{itemize}[<+->]
\item Determined by the overlap between two channels
\item One simplification: \myemph{the uncoupled closed-channel bound state, $\phi_{0}$, is much smaller than the interparticle distance in size}. 
\item Size of $\phi_{0}$ in the closed-channel: $\sim{E_{b}}^{-1/2}$
\item Size of atoms in the open-channel: interparticle distance $a_{0}\sim{}E_{F}^{-1/2}$
\item Controlling parameter
\begin{equation}\label{eq:pathInt2:zetaDef}
\boxed{\zeta=\frac{\Delta_{2}^{2}}{\Delta_{1}\eta}\sim\br{\frac{E_{F}}{E_{b}}}^{1/2}\ll1}
\end{equation}
\item Small $\zeta$ dictates a \myemph{perturbative} treatment. 
\end{itemize}
}

\frame{
\frametitle{Hamiltonian and Hubbard-Strotonovich trans.}
\begin{equation}
\begin{split}\label{eq:pathInt2:actionFermi}
&S(\bar\psi,\psi)=\int^{\beta}_{0}d\tau\int{d^{d}r}\\
&\Big[\sum_{j}\bar\psi_{j}(\partial_\tau-\nth{2m}\nabla^{2}-\mu+\eta_{j})\psi_{j}
-(\bar\psi\bar\psi)\tilde{U}(\psi\psi)\Big]
\end{split}
\end{equation}
Hubbard-Strotonovich transformation: introduce a two-component auxiliary field (order parameters)
\begin{equation*}
\mtrx{\Delta_{1}\\\Delta_{2}}\longrightarrow
	\mtrx{\Delta_{1}\\\Delta_{2}}-
	\mtrx{U&Y\\Y^{*}&V}
	\mtrx{\psi_{b}\psi_{a}\\\psi_{c}\psi_{a}}
\end{equation*}\pause
The action can then be rewritten in a more compact form with respect to $\Psi$ and $\bar\Psi$
\begin{equation}\label{eq:pathInt2:actionMixCompact}
S(\bar\Delta,\Delta,\bar\psi_{i},\psi_{i})=\int^{\beta}_{0}d\tau\int{d^{d}r}
	\mbr{\Delta^{\dg}\tilde{U}^{-1}\Delta-\bar\Psi\mathcal{G}^{-1}\Psi}
\end{equation}
%It is bilinear of $\Psi$.  We can integrate it out,
%\begin{equation}\label{eq:pathInt2:actionD}
%S(\bar{\Delta},\Delta)=\int{dx}\br{\bar{\Delta}\tilde{U}^{-1}\Delta-\tr\ln\nG}
%\end{equation}



}
\note{$\Delta_{1}$  $\Delta_{2}$ do not correspond to open-/closed-channel.}

\frame{
\frametitle{Mean-field equations}
 Integrate out $\psi_{i}$
\begin{equation}\label{eq:pathInt2:actionD}
S(\bar{\Delta},\Delta)=\int{dx}\br{\bar{\Delta}\tilde{U}^{-1}\Delta-\tr\ln\nG}
\end{equation}
Mean field equation:

  \begin{equation}\label{eq:pathInt2:mf}
\mtrx{\Delta_1\\\Delta_2}=\mtrx{U&Y\\Y^{*}&V}\sum_{\vk}\mtrx{h_{1\vk}\\h_{2\vk}}
\end{equation}
  where 
  \begin{gather}
  h_{1\vk}=\av{\psi_{a,-{\vk}}\psi_{b,+{\vk}}}
  =\Delta_{1}\frac{E_{1\,\vk}+\xi_{\vk}+\eta}{(E_{1\,\vk}+E_{2\,\vk})(E_{1\,\vk}+E_{3\,\vk})}\label{eq:pathInt2:h1}\\
  h_{2\vk}=\av{\psi_{a,-{\vk}}\psi_{c,+{\vk}}}
  =\Delta_{2}\frac{E_{1\,\vk}+\xi_{\vk}}{(E_{1\,\vk}+E_{2\,\vk})(E_{1\,\vk}+E_{3\,\vk})}\label{eq:pathInt2:h2}
  \end{gather}

}
\frame{
\frametitle{Order parameters}
\begin{itemize}

\item Gor'kov Green function for $\bar\Psi=\mtrx{\bar\psi_{a}&\psi_{b}&\psi_{c}}$, fermionic correlation
\begin{equation}\label{eq:pathInt2:nGDeltaK}
\mathcal{G}^{-1}=
\begin{pmatrix}
i\omega_{n}-\xi_{k}&\Delta_{1}&{\mathemph{\Delta_{2}}}\\
\bar\Delta_{1}&i\omega_{n}+\xi_{k}&0\\
\mathemph{\bar\Delta_{2}}&0&i\omega_{n}+\xi_{k}+\eta
\end{pmatrix}
\end{equation}
\item $\Delta_{1}$ includes the coupling with the closed-channel, $Y$; 

\item $\mathemph{\Delta_2}$ terms describe the inter-channel Pauli exclusion. 
\pause
\item Solved the three-species two-channel problem in two steps:
\begin{enumerate}
\item Treat effects other than the inter-channel Pauli exclusion in a non-perturbative way; ($\mathemph{\Delta_2=0}$)
\item Treat the inter-channel Pauli exclusion perturbavtively. 

\end{enumerate}
\end{itemize}

}

%\frame{
%\frametitle{Two-channel three-species model}
%\begin{itemize}
%\item Two order parameters 
% \begin{equation}\label{eq:pathInt2:mf}
%\mtrx{\Delta_1\\\Delta_2}=\mtrx{U&Y\\Y^{*}&V}\sum_{\vk}\mtrx{<{\psi_{a,-{\vk}}\psi_{b,+{\vk}}}>\\<{\psi_{a,-{\vk}}\psi_{c,+{\vk}}}>}
%\end{equation}
%\item Order parameters are result of mixed channel;
%\item $\Delta_{1}$ corresponds to the single channel gap;
%\item $\Delta_{1}$ includes the broad resonance limit effect (the $Y$ coupling with the closed-channel); 
%\item Reduced to the single-channel by two steps:
%\begin{enumerate}
%\item Treat effects other than the inter-channel Pauli exclusion in a non-perturbative way;
%\item Treat the inter-channel Pauli exclusion perturbavtively. 
%\end{enumerate}
%\end{itemize}
%}


\frame{
	\frametitle{Narrow resonance w/o interchannel Pauli exclusion}
	
	\begin{itemize}
\item Energy zero moves to the Fermi surface
\begin{gather}
1=-\mbr{\frac{4\pi{\mathemph{\tilde{a}_{s}}(\mu)}}{m}\sum(\nth{2E_{\vk}}-\nth{2\epsilon_{\vk}})}\label{eq:pathInt2:narrowGapS}\\
\tilde{a}_{s}=a_{\text{bg}}(1+\frac{\mathcal{K}}{\delta-\mathemph{2\mu}})\approx{}\frac{a_{\text{bg}}\mathcal{K}}{\delta-\mathemph{2\mu}}
\end{gather}
\item Extra counting for the closed-channel. 
\begin{gather}
\mathemph{N_{\text{open}}}=\sum_\vk\mbr{1-\frac{\epsilon_{\vk}}{E_{\vk}}}\label{eq:pathInt2:narrowNumS}
\end{gather}
\item  Consistent with  the study by Gurarie and Radzihovsky \footnote{[Gurarie and Radzihovsky, Annals of Physics, 322(1):2 � 119.  2007]}
\end{itemize}
}
\frame{
\frametitle{Narrow resonance w/o interchannel Pauli exclusion}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.8\textwidth]{narrow}
%\caption{The chemical potential $\mu$, the open-channel gap $\Delta_{1}$ and the open-channel relative weight $\beta_{o}$ over crossover with the narrow Feshbach resonance without considering the inter-channel Pauli exclusion  vs. the chemical potential and the gap in the single-channel model} 
\label{fig:pathInt2:narrow}
%{\small  The gap in the open-channel $\Delta_{1}$ and the chemical potential $\mu$ are rescaled with the the Fermi energy of the total density, $E_{F}^{(tot)}$.  The $x$-axis is the detuning $\delta$ rescaled with $E_{0}$ (see Eq. \ref{eq:pathInt2:E0}) for the narrow resonance; while it is $-1/a_{s}k_{F}$ for the single-channel curves. We have taken $\delta_{c}=0.001E_{F}^{(tot)}$ for the narrow resonance figure. We used the detuning from the resonant point  for the $x$-axis in the narrow resonance instead of $-1/a_{s}k_{F}$ in the single-channel because the additional shift, $2\mu$, considering in Eq. \ref{eq:pathInt2:simplenarrowAs}.  Consequently, the chemical potential lines in both cases cross the $x$-axis at the same point where $\mu=0$.  }
\end{center}
\end{figure}
\begin{itemize}
\item $\Delta_{1}$ saturates in BEC side
\item $\tilde{a}_{s}$ differs from two-body $a_{s}$ with special shift $2\mu$
\item $\beta_{o}=\frac{n_{open}}{n_{total}}$ deviates from 1 substantially at universality

\end{itemize}
}




\frame{
\frametitle{Fermionic modes}
Diagonalize Gor'kov Greens function
\begin{columns}
\column{0.55\textwidth}
\begin{align*}
E_{1\vk}&\approx{}E_{\vk}+\mathemph{u_{\vk}^{2}\Delta_{1}\zeta}\\
E_{2\vk}&\approx{}E_{\vk}-\mathemph{v_{\vk}^{2}\Delta_{1}\zeta}\\
E_{3\vk}&\approx{}\epsilon_{\vk}+\eta+\mathemph{\frac{\zeta}{2}\Delta_{1}}
\end{align*}
\column{0.45\textwidth}
\begin{equation*}
\boxed{\zeta=\frac{\Delta_{2}^{2}}{\Delta_{1}\eta}}
\end{equation*}
$E_\vk=\sqrt{(\epsilon_\vk-\mu)^2+\abs{\Delta_1}^2}$
\end{columns}
\begin{itemize}
\item $E_{1\vk}$ and $E_{2\vk}$ are like Bogoliubov quasi-paticle; 
\item $E_{1\vk}$ and $E_{2\vk}$ now split by $\Delta_1\zeta$;
\item $E_{3\vk}$ is the free fermionic excitation in the close-channel;
\pause
\item The correction is in the order of $\zeta\ll1$;
\item The correction is due to the inter-channel Pauli exclusion and does not disappear even at the extremely narrow limit. 
\end{itemize}

}



\frame{
\frametitle{Bosonic modes}
Fluctuations of the order parameters;
\begin{itemize}
\item<1-> Order parameters describe the collective properties of atoms;
\item<2-> Four possible modes for two complex order parameters ($\Delta_1$, $\Delta_2$);
\item<3-> The overall phase fluctuation mode is the Goldstone mode;
\item<3-> The other three modes are massive;
\item<4-> The inter-channel out-of-sync phase mode is gapped at pair-breaking energy;
\item<5-> Corrections due to the inter-channel Pauli exclusion is in the order of $\zeta$.
\end{itemize}
}

\section{Discussion}
\frame{
\frametitle{Discussion}
\begin{itemize}
\item<1-> The inter-channel Pauli exclusion can be handled by perturbation method;
\item<1-> Handling two-channel problem, especially the inter-channel Pauli exclusion effect, self-consistently;
\item<2-> Using the BCS-approach (non-perturbative) as the zeroth order, building narrow resonance with the  inter-channel Pauli exclusion upon it;
\item<2-> The methodology about narrow resonance is still valid for non-simple-BCS treatment about crossover;

\item<3-> Finite temperature: BCS $\Rightarrow$ Fermi liquid; BEC $\Rightarrow$ normal dimer gas.  
\end{itemize}
}
\frame{
\begin{center}
 \Huge Thank you!
\end{center}

}
\end{document}

